BIFURCATIONS OF APPROXIMATE HARMONIC BALANCE SOLUTIONS AND TRANSITION TO CHAOS IN AN OSCILLATOR WITH INERTIAL AND ELASTIC SYMMETRIC NONLINEARITIES

The local stability of approximate periodic solutions and period-doubling bifurcations in a harmonically forced non-linear oscillator with symmetric elastic and inertia non-linearities are studied analytically and numerically. Approximate principal resonance solutions are first obtained using a two-term harmonic balance and then a consistent second order stability analysis of the associated linearized variational equation is carried out using approximate methods to predict zones of symmetry breaking leading to period-doubling bifurcation and chaos. The results of the present work, which follows the analysis approach presented by Szemplinska-Stupnika (1986 International Journal of Nonlinear Mechanics23, 257–277; 1987 Journal of Sound and Vibration113, 155–172) are verified for selected system parameters by numerical simulations using methods of qualitative theory, and good agreement was obtained between the analytical and numerical results. Finally, a criterion for the period-doubling bifurcation is proposed analytically, for this type of oscillator, and compared with computer simulation results that predict the true period-doubling bifurcation and chaos boundaries.